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The great mathematician G. H. Hardy told us that "Beauty is the ?rst test: there is no permanent place in the world for ugly mathematics" (see [24, p. 85]). It is clear why Hardy loved complex analysis: it is a very beautiful partofclassicalmathematics. ThetheoryofHilbertspacesandofoperatorson themisalmostasclassicalandisperhapsasbeautifulascomplexanalysis. The studyoftheHardy-Hilbertspace(aHilbertspacewhoseelementsareanalytic functions), and of operators on that space, combines these two subjects. The interplay produces a number of extraordinarily elegant results. For example, very elementary concepts from Hilbert space provide simple proofs of the Poisson integral (Theorem 1. 1. 21 below) and Cauchy integral (Theorem 1. 1. 19) formulas. The fundamental theorem about zeros of fu- tions in the Hardy-Hilbert space (Corollary 2. 4. 10) is the central ingredient of a beautiful proof that every continuous function on [0,1] can be uniformly approximated by polynomials with prime exponents (Corollary 2. 5. 3). The Hardy-Hilbert space context is necessary to understand the structure of the invariant subspaces of the unilateral shift (Theorem 2. 2. 12). Conversely, pr- erties of the unilateral shift operator are useful in obtaining results on f- torizations of analytic functions (e. g. , Theorem 2. 3. 4) and on other aspects of analytic functions (e. g. , Theorem 2. 3. 3). The study of Toeplitz operators on the Hardy-Hilbert space is the most natural way of deriving many of the properties of classical Toeplitz mat- ces (e. g. , Theorem 3. 3.
List of contents
The Unilateral Shift and Factorization of Functions.- Toeplitz Operators.- Hankel Operators.- Composition Operators.- Further Reading.
Summary
This book offers an elementary and engaging introduction to operator theory on the Hardy-Hilbert space. It provides a firm foundation for the study of all spaces of analytic functions and of the operators on them. Blending techniques from "soft" and "hard" analysis, the book contains clear and beautiful proofs. There are numerous exercises at the end of each chapter, along with a brief guide for further study which includes references to applications to topics in engineering.
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From the reviews:
"This text appears as number 237 in the Graduate Texts in Mathematics series published by Springer. … possible textbook for graduate courses, as well as independent study and reference for research. The book is very well written and contains very elegant proofs. … Each chapter in the book has plenty of exercises … . In short: this is a wonderful book, a pleasure to read and use as a text, or add to any mathematician’s ‘collection’ of references to a beautiful and rich subject." (Mihaela Poplicher, MathDL, April, 2007)
"The aim of this book is to provide an introduction to operator theory on the Hardy space H
2, also called the Hardy-Hilbert space. … Each chapter ends with a list of exercises and notes and remarks. … The book gives an elementary and brief account of some basic aspects of operators on H
2 and can be used as a first introduction to this area." (Takahiko Nakazi, Mathematical Reviews, 2007 k)
"This text is a gentle introduction to the ‘concrete’ operator theory on the Hardy-Hilbert space H
2, an important model space for operator theory on function spaces. … Each chapter contains a number of exercises. … Historical notes and remarks are also included. The bibliography is exhaustive and up to date. The book seems completely suitable for a first- or second-year graduate course in function spaces and operator theory … and can probably be adapted to both a basic and a more advanced course." (Dragan Vukotic, Zentralblatt MATH, Vol. 1116 (18), 2007)
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From the reviews:
"This text appears as number 237 in the Graduate Texts in Mathematics series published by Springer. ... possible textbook for graduate courses, as well as independent study and reference for research. The book is very well written and contains very elegant proofs. ... Each chapter in the book has plenty of exercises ... . In short: this is a wonderful book, a pleasure to read and use as a text, or add to any mathematician's 'collection' of references to a beautiful and rich subject." (Mihaela Poplicher, MathDL, April, 2007)
"The aim of this book is to provide an introduction to operator theory on the Hardy space H2, also called the Hardy-Hilbert space. ... Each chapter ends with a list of exercises and notes and remarks. ... The book gives an elementary and brief account of some basic aspects of operators on H2 and can be used as a first introduction to this area." (Takahiko Nakazi, Mathematical Reviews, 2007 k)
"This text is a gentle introduction to the 'concrete' operator theory on the Hardy-Hilbert space H2, an important model space for operator theory on function spaces. ... Each chapter contains a number of exercises. ... Historical notes and remarks are also included. The bibliography is exhaustive and up to date. The book seems completely suitable for a first- or second-year graduate course in function spaces and operator theory ... and can probably be adapted to both a basic and a more advanced course." (Dragan Vukotic, Zentralblatt MATH, Vol. 1116 (18), 2007)
